SIS Epidemic Model

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The SIS Epidemic Model is a way of modeling diseases by classifying the population based on their disease status. In general, these types of models are referred to as SIR models, referring to Susceptible, Infected and Removed/Recovered as the three disease statuses the population can fall into.

  • S - The susceptible population who are not affected but are at risk for infection.
  • I - Infected individuals who are capable of transmitting the disease
  • R - Individuals who have either recovered and gained permanent immunity from the disease, or are otherwise removed from the population and can no longer infect susceptible individuals (death, quarantine).

Based on these classifications, we can see the SIS model is one which models diseases in which recovered individuals are not granted immunity to the disease and are again susceptible to infection upon recovery. The SIS model can be used to describe sexually transmitted diseases such as syphilis or gonorrhea in which individuals are treated and recover but are immediately susceptible again.

This simple version of the model assumes a closed population; no one is born and no one dies, so the population is constant and every individual is either part of S or part of I. A general case of the model has the following form:

    `(dS)/(dt) = - (beta)/N * SI+gamma*I`
    `(dI)/(dt) =(beta)/N*SI-gamma*I`

In this model, `beta` is the average number of disease-spreading contacts made by each infected individual in unit time, `N` is the population, and `gamma` is the recovery rate (thus, `1/gamma` is the average infectious period). Notice that `(dS)/(dt)=-(dI)/(dt)` or `(dS)/(dt)+(dI)/(dt)=0`, so we can see again that the total population is unchanging regardless of time.

If `beta>gamma` - that is, if infected individuals infect others faster than they recover - the infected population approaches `((beta-gamma)N)/beta` and the susceptible population approaches `(gamma*N)/beta`, and the disease remains endemic. If `beta<gamma` then the infected population are recovering faster than they can infect the susceptible population, so the infected population approaches zero and the epidemic ends.

To see why this is the case, consider the ratio `beta/gamma`. This is referred to as the basic reproduction number `mathcalR_0`. Since `beta` is the number of "successful" contacts by each infected individual in unit time, and `1/gamma` is the average infectious period (or average time time an individual stays infected),  `mathcalR_0` is the average number of susceptible individuals one infected individual will infect during the course of their infection. So we can see that if `mathcalR_0 >1` then as we said above, the infected individuals are infecting the susceptible population faster than they're recovering, and if `mathcalR_0<1` then they are not.  In the case of the SIS model in which recovered individuals are not immune, `mathcalR_0>1` means that some part of the population will always be infected, so we say the disease becomes endemic.

Reference

Allen, Linda J. S. "6.8.1 SI, SIS, and SIR Epidemic Models." An Introduction to Mathematical Biology. Upper Saddle River, NJ: Pearson/Prentice Hall, 2007. 271-73. Print.

 

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